Question

If ${{2}^{x}}-{{2}^{x-1}}=4$, then the value of ${{x}^{x}}$ is
(a) 7
(b) 3
(c) 27
(d) None of these

Answer 1

Hint: Simplify the given equation by taking out the common terms and using laws of exponents that state that ${{a}^{b}}\times {{a}^{c}}={{a}^{b+c}}$. Calculate the value of ‘x’ which satisfies the given equation. Substitute the value of ‘x’ in the given expression and calculate its value.

Complete Step-by-step answer:
We have the equation ${{2}^{x}}-{{2}^{x-1}}=4$. We have to calculate the value of ${{x}^{x}}$.
We will use the law of exponents to simplify the given equation.
We know that the laws of exponents state that ${{a}^{b}}\times {{a}^{c}}={{a}^{b+c}}$.
So, we can rewrite ${{2}^{x-1}}$ as ${{2}^{x-1}}={{2}^{x}}\times {{2}^{-1}}$.
Thus, we can rewrite the equation ${{2}^{x}}-{{2}^{x-1}}=4$ as \[{{2}^{x}}-{{2}^{x}}\times {{2}^{-1}}=4\].
Taking out the common terms in the above equation, we have ${{2}^{x}}\left( 1-{{2}^{-1}} \right)=4$.
Simplifying the above equation, we have ${{2}^{x}}\left( 1-{{2}^{-1}} \right)={{2}^{x}}\left( 1-\dfrac{1}{2} \right)={{2}^{x}}\dfrac{1}{2}=4$.
Rearranging the terms of the above equation, we have ${{2}^{x}}=4\times 2=8$.
We can rewrite the above equation as ${{2}^{x}}={{2}^{3}}=8$.
Thus, we have $x=3$.
We will now calculate the value of the expression ${{x}^{x}}$.
Substituting $x=3$ in the expression ${{x}^{x}}$, we have ${{x}^{x}}={{3}^{3}}$.
Thus, we have ${{x}^{x}}={{3}^{3}}=27$.
Hence, the value of the expression ${{x}^{x}}$ is 27, which is option (c).

Note: We can’t solve this question without using the laws of exponents. We observe that ${{x}^{x}}$ is an algebraic expression. However, it acquires a fixed value as $x=3$ is the solution of the equation ${{2}^{x}}-{{2}^{x-1}}=4$.